Refining Diffusion Approximations for Queues

Diffusion approximations and associated heavy-traffic Emit theorems have now become basic tools of queuing theory; e.g., see Gelenbe and Mitrani [3, Chapter 4], Kleinrock [8, Chapter 2]," Newell [12, Chapter 6] and references there. Under heavy-traffic condiuons, diffusion approximations are identified and justified by the heavy-traffic limit theorems: the diffusion process is obtained as the limit. Variants of the limiting diffusion process also may be useful as approximations in moderate traffic. However, in moderate traffic, it is often difficult to select an appropriate diffusion process. Whenever possible, it is important to consider refining the diffusion approximations, for example, by comparing the approximations with available bounds and exact resuits. The purpose of this note is to illustrate the need to refine diffusion approximations by considering diffusion approximations in one of the simplest settings: approximations for the expected waiting time in the standard G I / G / I queue. In the G I / G / I model there is a single server, unlimited waiting space, the first-come-first-served discipline and independent sequences of independent and identically distributed interarrival times and service times. Perhaps the most important insight provided by the heavy-traffic limit theorems for the G I / G / I queue is the fact that only the first two moments of the interarrival-time and service-time distributions are essential for determining the standard congestion measures in heavy traffic (when the traffic intensity is close to its critical value). However, one should not rely too heavily on the associated approximations because different approximations can be obtained by very similar reasoning. In Section 2, we apply a standard heuristic argument to obtain diffusion approximations for three basic processes in the G I / G / ! queue: the sequence of successive waiting times, (W~, n ~ O}, the virtual waiting time process, {V(t), t ~ 0), and the queue length process, (Q(t), t>_.0). In each case, the diffusion process approximation for the queuing process yields an approximation for the mean of the equilibrium distribution. However, the three means are related exactly so that we have three different ways to obtain ak~ approximation for the expected waiting time. To specify the connecting relationships, we introduce our notation. Let ~ be the arrival rate,/t the service rate, p ffi k/p the traffic intensity, ca 2 the squared coefficient of variation of the interarrival-time distribution (variance divided by the square of the mean), and c, 2 the squared coefficient of variation of the service-time distribution. Let W, V and Q be random variables with the equilibrium distributions of Wn, V(t) and Q(t), respectively. To go from EQ to E W, we apply Little's for-